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Published in Philip A. Laplante, Comprehensive Dictionary of Electrical Engineering, 2018
where the frequency u is expressed in radians. See Fourier transform. Fourier optics optical systems that utilize the exact Fourier transforming properties of a lens. Fourier optics relay lens a lens system that produces the exact Fourier transform of an image. Two such relay lens will reproduce an image without any phase curvature. Fourier phase the phase angle (modulo 2) taken by the Fourier transform. Specifically, given the Fourier transform F( f ) = A( f )(u)ei ( f )(u) , then ( f ) represents the phase. See Fourier transform. Fourier phase congruence for a 1-D realvalued signal f and a point p, the Fourier phase that the signal f would have if the origin were shifted to p; in other words, it is the Fourier phase of f translated by - p. The congruence between the phases at p for the various frequencies -- in other words the degree by which those phases at p are close to each other -- can be measured by f ( p) where f ( p)2 + H ( f )( p)2 , where H ( f ) is the Hilbert transform of f . See Fourier amplitude/ phase, Hilbert transform. Fourier plane a plane in an optical system where the exact Fourier transform of an input image is generated. Fourier series Let f (t) be a continuous time periodic signal with fundamental period T such
Guided-Wave Fourier Optics
Published in Shyamal Bhadra, Ajoy Ghatak, Guided Wave Optics and Photonic Devices, 2017
Fourier transform offers the orthogonality needed for such an operation due to the summation of the harmonic terms of sine and cosine terms. Thus, Fourier optics offers an excellent technique for processing optical signals. Originally, the idea of transformation in optics was investigated some decades ago. The Fourier transform of a continuous and coherent spatial distribution or image can be evaluated physically to a high degree of accuracy by using one or more simple lenses plus free-space light propagation, leading to the well-established technology of Fourier optics as described in texts by Goodman [7], Papoulis [8] and Gaskill [9]. This can be applied to a sequence of modulated lightwave signals carried by several optical subcarriers so that these channels can be positioned to be orthogonal to each other. This is one of the main features of the DFT.
Fourier Guided Wave Optics
Published in Le Nguyen Binh, Wireless And Guided Wave Electromagnetics, 2017
Fourier transform offers the orthogonality needed for such an operation due to the summation of the harmonic terms of sine and cosine terms. Fourier optics offers an excellent technique for processing optical signals. The idea of transformation in optics was investigated some decades ago. The Fourier transform of a continuous and coherent spatial distribution or image can be evaluated physically to a high degree of accuracy by use of one or more simple lenses plus free-space light propagation, leading to the well-established technology of Fourier optics as described in texts by Goodman,8 Papoulis,9 and Gaskill.10 This can be applied to a sequence of modulated lightwave signals carried by several optical subcarriers so that these channels can be positioned to be orthogonal to each other. This is one of the main features of the discrete Fourier transform (DFT).
Lensless broadband diffractive imaging with improved depth of focus: wavefront modulation by multilevel phase masks
Published in Journal of Modern Optics, 2019
Vladimir Katkovnik, Mykola Ponomarenko, Karen Egiazarian
In Fourier optics, the incoherent image formation is modelled as a convolution of the true object image and the PSF of the system (12). Let us present this formalism in order to introduce a notation and to discuss some assumptions of this modelling.
The existence and evolution of fast-decaying Bessel modes in cylindrical hollow waveguides and in free space
Published in Journal of Modern Optics, 2019
G. Nyitray, A. Nagyváradi, M. Koniorczyk
The basic idea of Fourier optics is that beams and pulses can be described as superpositions of plane waves which travel in different directions. This explains why most beams and pulses exhibit diffraction, and consequently an increasing spatial broadening during their propagation. Surprisingly, however, there exist solutions to the scalar and vectorial wave equations in homogeneous media, which resist the effect of diffraction for long distances both in free space [1–3], and waveguides [4–8]. Propagation invariant Bessel beams are exact solutions of the wave equation, best expressed in a cylindrical coordinate system. The propagation of continuous beams with radial symmetry is described by the Helmholtz equation where is a travelling wave solution for this equation. Harmonic time dependence has been assumed in the wave equation, with ω the light frequency, the wave number and c the light velocity in the medium. It is worth mentioning that several propagation invariant solutions of the wave equation are also known, each suitable for description in a given coordinate system type, for instance, cosine fields in Cartesian coordinates, Mathieu beams in elliptic coordinates and parabolic beams in parabolic coordinates [3]. As we have mentioned, Bessel beams are eigenmodes of the Helmholtz equation in a cylindrical orthogonal coordinate system, moreover they can possibly satisfy Maxwell's equations as well. The Helmholtz equation can be derived from Maxwell's equations thus the solutions of Maxwell's equations obey the wave equations. Conversely, Maxwell's equations cannot be derived from the wave equations; hence any particular solution of the wave equation does not necessarily satisfy Maxwell's equations. In Ref. [9] it was pointed out that the wave equation could be applied to the field or to a Hertzian vector potential. If we chose the Hertz vector proportional to the electric and magnetic components of the simplest propagation invariant wave which satisfy Maxwell's equations can be expressed by zeroth-order Bessel function as follows: where is the field amplitude, ρ is the radial coordinate, is the n'th root of , a is the radius of the waveguide, and the magnitude of the transverse and longitudinal components of the wavenumber, respectively. is the derivative of the function, Ω is the wave impedance of the vacuum (free space). The details of the derivation of the vectorial field components from the Hertz vector are given in Section 5.